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@MISC{Boumans_fisher’sinstrumental, author = {Marcel Boumans}, title = {Fisher’s Instrumental Approach to Index Numbers. Supplement to the History of Political Economy}, year = {}}

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Abstract

‘If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. ’ (Hilbert 1902, 448) ‘Sometimes control with a single lens is impossible since some incompatible features are required and a compromise becomes necessary calling for further judgement on the part of the designer as to which error should be reduced and to what degree. ’ (Bracey 1960, 18) 1.

...ther reason, crucial for the microeconomic approach, is that “so long as we stick to the economic theory of index numbers, the circular test is as required as is the property of transitivity itself” (=-=Samuelson and Swamy 1974-=-, 576). Moreover, the forms of the indexes Samuelson and Swamy provided contradicted Fisher’s assertion that only fixed weights lead to satisfaction of the Circular Test. The inconsistencies between t...

...rong belief in economics that axiomatization leads to better theories. One finds the same kind of conviction in measurement theory – a standard work in measurement theory, Foundations of Measurement (=-=Krantz et al. 1971-=-) is exemplary for that conviction. But the axiomatic approach is also dominant in index number theory. Whether this belief is justified or not, even the most authoritarian axiomatizer of economics, J...

...on, satisfying a number of axioms. These axioms do not determine a unique form of the price index function. Several additional tests are needed ‘for assessing the quality of a potential price index’ (=-=Eichhorn & Voeller 1976-=-, 29). Both axioms and tests are formalized as functional equations. Then the inconsistency theorems can be proven by showing that for the relevant combinations of functional equations the solution sp...

...btained.sThe reasons why there is any such thing as 'map-projection problem' is 'the fact that every map, large or small, must have some distortions and that every world map must have interruptions' (=-=Fisher & Miller 1944-=-, 3). To flatten out a globe, unavoidably you have to stretch and/or shrink it in certain directions and you have to tear it at several places. In particular, there is a tension between interruption a...

... ‘de-empirization’, as axiomatization was called by him: ‘at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is insdanger of degeneration’ (von =-=Neumann 1961-=-, 9). According to von Neumann, the ‘prime’ reason, why, of all Euclid’s postulates, the fifth was questioned, was its unempirical character (idem, 3). He found it ‘hardly possible to believe in the e...

... and The Making of Index Numbers 1922. In these, Fisher evaluated index formulae in a systematic way, with respect to a number of ‘tests’. Although they are considered as the ‘Old and New Testament’ (=-=Vogt and Barta 1997-=-, viii) of Axiomatic Index Theory, the axiomatic approach originated from challenges to Fisher’s system of tests on grounds of their consistency and the seeming arbitrariness of the choice of tests. T...

...well informed about this development. In 1890 and 1891, in addition to the courses he was taking in graduate school, he started teaching geometry at Yale (which he continued to do for several years) (=-=Allen 1993-=-, 36). He even wrote a textbook on geometry, Elements of Geometry (1896), with Andrew W. Phillips of the Mathematics Department. Fisher also took courses under Josiah Willard Gibbs, one of his mentors...

...matic formulation of measurable utility. ‘Though this approach is so familiar today, after the work of von Neumann and Morgenstern, Frisch’s paper is very possibly the first formulation of its type’ (=-=Arrow 1960-=-, 176). It is considered as a classic in the theory of consumer behavior because it, ‘apparently’ for the first time, introduced the axiomatic approach into the theory ofseconomic choice (Chipman et a...

...le since some incompatible features are required and a compromise becomes necessary calling for further judgement on the part of the designer as to which error should be reduced and to what degree.’ (=-=Bracey 1960-=-, 18) 1. Introduction Since the beginning of the 19th century a large number of price index number formulae have been developed, mostly named after their inventors, such as the Paasche and Layspeyres ...

...s apparently destroyed for him the usefulness of the model. Accordingly he usuallyshad recourse to the geometrical representation of his equations, and this method he used with great ease and power’ (=-=Bumstead 1961-=-, xii-xiii). While Gibbs saw geometrical illustrations mainly as aids to the imagination, Fisher stressed the role of visualizations because they helped to understand a system or phenomenon. It connec...

... them. ‘What appears to be missing, is a codification of the rules which should define and guide the use of mathematics as an instrument for the description, interpretation and control of phenomena’ (=-=Israel 1981-=-, 219). As we shall see, the rule implied by Fisher’s instrumental approach can be well described as finding the best balance between theoretical and empirical requirements, even if these requirements...

...Frisch's result that Fisher's tests are inconsistent was never doubted but his proofs were (unjustly) criticized which created some confusion about Frisch’s results. In a debate that took many years (=-=Swamy 1934-=-, Frisch 1934, Swamy 1940, Swamy 1965), both the correctness of Frisch's proofs and the economic interpretation and significance of Fisher's tests were discussed. The inconsistency of the tests called...

...st (F7-C) and the Factor Reversal Test (F9). He saw the Factor Reversal Test as ‘economically completely unfounded’ (‘ökonomisch vollkommen unbegründet’), that is, its economic meaning was not given (=-=Wald 1937-=-, 183). He concluded that the formal-mathematical way was not well suited to the solution of the index problem, so the larger part of his paper was given to a discussion of the economic approach to in...

...these qualities can be obtained. The “map-projection problem” exists because of “the fact that every map, large or small, must have some distortions and that every world map must have interruptions” (=-=Fisher and Miller 1944-=-, 3). To flatten out a globe, one must stretch and/or shrink it in certain directions and tear it at several places. In particular, there is a tension between interruption and distortion: only by incr...